RTT relations, a modified braid equation and noncommutative planes

With the known group relations for the elements $(a,b,c,d)$ of a quantum matrix $T$ as input a general solution of the $RTT$ relations is sought without imposing the Yang–Baxter (YB) constraint for $R$ or the braid equation for $R̂=PR.$ For three biparametric deformatios, $GL(p,q)(2),$ $GL(g,h)(2),$ and $GL(q,h)(1/1),$ the standard, the nonstandard, and the hybrid one, respectively, $R$ or $R̂$ is found to depend, apart from the two parameters defining the deformation in question, on an extra free parameter $K,$ such that $R̂(12)R̂(23)R̂(12)−R̂(23)R̂(12)R̂(23)=[(K/K1)−1][(K/K2)−1](R̂(23)−R̂(12))$ with $(K1,K2)=(1,p/q),(1,1),$ and $(1,1/q),$ respectively. Only for $K=K1$ or $K=K2$ one has the braid equation. Arbitray $K$ corresponds to a class (conserving the group relations independent of $K)$ of the MQYBE or modified quantum YB equations studied by Gerstenhaber, Giaquinto, and Schack. Various properties of the triparametric $R̂(K;p,q),$ $R̂(K;g,h),$ and $R̂(K;q,h)$ are studied. In the larger space of the modified braid equation (MBE) even $R̂(K;p,q)$ can satisfy $R̂2=1$ outside the braid equation (BE) subspace. A generalized, $K$-dependent, Hecke condition is satisfied by each three-parameter $R̂.$ The role of $K$ in noncommutative geometries of the $(K;p,q),$ $(K;g,h),$ and $(K;q,h)$ deformed planes is studied. $K$ is found to introduce a “soft symmetry breaking,” preserving most interesting properties and leading to new interesting ones. Further aspects to be explored are indicated.